In this talk we will be considering corotational, energy-supercritical wave maps from Minkowski space into target manifolds which are geometrically close to a sphere. It is already known that if the target manifold is precisely a sphere, the wave maps model exhibits explicitly known self-similar blowup solutions, which are asymptotically nonlinear stable against small perturbations. Using perturbative methods we show that there do not only exist self-similar solutions to the wave maps equation when the target manifold is a perturbed sphere but that they are also asymptotically nonlinear stable.